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Lecture #8 of 24Study skills workshop – Cramer 121 – 7-9 pm Prof. Bob Cormack – The illusion of understanding

Homework #3 – Moment of Inertia of disk w/ hole Buoyancy

Energy and Conservative forces Force as gradient of potential Curl

Stokes Theorem and Gauss’s Theorem Line Integrals Energy conservation problems Demo – Energy Conservation

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2

Disk with Hole

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2Parallel CM totalI I M d

Axis 1 (through CM)

Axis 2 (Parallel to axis 1)

d

R, M

3

Buoyancy

Archimedes Taking bath Noticing water displaced by his body as he got

in the tub Running starkers thru the streets shouting “Eureka”

The buoyant force on an object is equal to the WEIGHT of the fluid displaced by that object

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Impure crown?King Hiero commissions

”Mission oriented Research”

bF Vg

4

Force as the gradient of potential

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00( ) ( ) ( ) ( )

( ) ( )

x r

r

r rx x

Scalar Vector

F x f x dx U r F r dr

dFf x F r U

dx

ˆ ˆ ˆ

1 1ˆ ˆˆsin

x y zx y z

rr r r

3 sinU Axy B Cz

5

Gravitational Potential

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ˆ ˆ ˆ

1 1ˆ ˆˆsin

x y zx y z

rr r r

2

( )

1

1

0

r

GMmU r

r

F U GMmr

GMmF GMm

r r r

F F

6

Curl as limit of tiny line-integrals

:25

0ˆ( ) lim

ia i

F drcurl F n

a

7

Stokes and Gauss’s theorem’s

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Gauss – Integrating divergence over a volume is equivalent to integrating function over a surface enclosing that volume.

Stokes – Integrating curl over an area is equivalent to integrating function around a path enclosing that area.

8

The curl-o-meter (by Ronco®)

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Conservative force

0 0F dr F

a

c d

b

e fˆ ˆ ˆ

det

x y z

x y z

fx y z

f f f

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Maxwell’s Equations

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0

0 0 0

; 0E B

BE

t

EB J

t

0B IA

Curl is zero EXCEPTWhere there is a

current I.

10

L8-1 – Area integral of curl

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Taylor 4.3 (modified)

ˆ ˆ( )F r yx xy

O

y

xP(1,0)

Q(0,1)

a

c

b

Calculate, along a,c

Calculate, along a,b

Calculate, inside a,c

Calculate, inside a,b

OQPF dr

OQP

F dA

11

L8-2 Energy problems

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A block of mass “m” starts from rest and slides down a ramp of height “h” and angle “theta”.

a) Calculate velocity “v” at bottom of ramp

b)Do the same for a rolling disk (mass “m”, radius r)

c) Do part “a” again, in the presence of friction “”

O

y

x

m

h

m

y

x

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Retarding forces summary

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1 ;gt

vzv v

gv

be

m

2 2 2ˆ ˆ16dragF D v v cv v

Linear Drag on a sphere (Stokes)

Falling from rest w/ gravity

Quadratic Drag on aSphere (Newton)

Falling from rest w/ gravity

Decelerating from v without gravity

ˆ ˆ3dragF D ux bux

2 mgv

c

( ) tanh( / )v t v gt v

0

0

( )1

vv t

cvt

m

13

Lecture #8 Wind-up

.

Read Chapter 49/24 Office hours 4-6 pmFirst test 9/26HW posted mid-afternoon

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Stokes’ Theorem

For conservative forces

( )Path SurfaceF dr F dA

0F